Question

two parallel lines, m and n, are cut by the transversal b as shown. suppose m∠1 = 65°. complete the statements below. we see that ∠1 and ∠2 are (choose one) thus, ∠1 and ∠2 are (choose one) so, m∠2 = □°. we see that ∠2 and ∠3 are (choose one) and since the lines m and n are parallel, ∠2 and ∠3 are (choose one) so, m∠3 = □°. therefore, ∠1 and ∠3 are (choose one) we also see that ∠1 and ∠3 are (choose one) the relationship between ∠1 and ∠3 is an example of the following rule. when parallel lines are cut by a transversal, (choose one)

Explanation:

Step1: Identify the relationship between ∠1 and ∠2

∠1 and ∠2 are vertical angles.
Vertical angles are congruent.
Since \(m\angle1 = 65^{\circ}\), then \(m\angle2=65^{\circ}\)

Step2: Identify the relationship between ∠2 and ∠3

∠2 and ∠3 are alternate interior angles.
When two parallel lines are cut by a transversal, alternate interior angles are congruent.
So \(m\angle3 = 65^{\circ}\)

Step3: Identify the relationship between ∠1 and ∠3

∠1 and ∠3 are congruent.
∠1 and ∠3 are also equal in measure.
The relationship between ∠1 and ∠3 is an example of the rule: when parallel lines are cut by a transversal, vertical angles (∠1 and ∠2) and alternate - interior angles (∠2 and ∠3) lead to non - adjacent angles (∠1 and ∠3) being congruent.

Answer:

We see that \(∠1\) and \(∠2\) are (vertical angles).
Thus, \(∠1\) and \(∠2\) are (congruent).
So, \(m∠2 = 65^{\circ}\).
We see that \(∠2\) and \(∠3\) are (alternate interior angles).
And since the lines \(m\) and \(n\) are parallel, \(∠2\) and \(∠3\) are (congruent).
So, \(m∠3 = 65^{\circ}\).
Therefore, \(∠1\) and \(∠3\) are (congruent).
We also see that \(∠1\) and \(∠3\) are (equal in measure).
The relationship between \(∠1\) and \(∠3\) is an example of the following rule. When parallel lines are cut by a transversal, (non - adjacent angles can be congruent through a series of angle - relationship properties: vertical angles and alternate - interior angles).